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Given a norm, one can evaluate both sides of the parallelogram law above. A remarkable fact is that if the parallelogram law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the p {\displaystyle p} -norm if and only if p = 2 , {\displaystyle p=2,} the so-called Euclidean norm or standard norm.
The base pairs form a parallelogram with half the area of the quadrilateral, A q, as the sum of the areas of the four large triangles, A l is 2 A q (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, A s is a quarter of A l (half linear dimensions yields quarter area), and the area of the parallelogram is A ...
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
Parallelogram law – Sum of the squares of all 4 sides of a parallelogram equals that of the 2 diagonals; Polarization identity – Formula relating the norm and the inner product in a inner product space; Ptolemy – Roman astronomer and geographer (c. 100–170) Ptolemy's table of chords – 2nd century AD trigonometric table
Rhomboid: a parallelogram in which adjacent sides are of unequal lengths, and some angles are oblique (equiv., having no right angles). Informally: "a pushed-over oblong". Not all references agree; some define a rhomboid as a parallelogram that is not a rhombus. [4] Rectangle: all four angles are right angles (equiangular). An equivalent ...
The sum of the distances from any interior point of a parallelogram to the sides is independent of the location of the point. The converse also holds: If the sum of the distances from a point in the interior of a quadrilateral to the sides is independent of the location of the point, then the quadrilateral is a parallelogram. [3]
For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six ...
The first property implies that every rhombus is a parallelogram. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel; adjacent angles are supplementary; the two diagonals bisect one another; any line through the midpoint bisects the area; and the sum of the squares of the sides equals the ...